{"id":1287,"date":"2025-10-17T13:14:40","date_gmt":"2025-10-17T13:14:40","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=1287"},"modified":"2025-11-03T15:17:30","modified_gmt":"2025-11-03T15:17:30","slug":"center-of-mass-and-moments-of-inertia-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/center-of-mass-and-moments-of-inertia-learn-it-3\/","title":{"raw":"Center of Mass and Moments of Inertia: Learn It 3","rendered":"Center of Mass and Moments of Inertia: Learn It 3"},"content":{"raw":"<h2>Center of Mass and Moments of Inertia in Three Dimensions<\/h2>\r\nAll the formulas we developed for double integrals can be extended to triple integrals for three-dimensional objects.\r\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Mass of a Three-Dimensional Solid<\/h3>\r\n<p class=\"whitespace-normal break-words\">For a solid object [latex]Q[\/latex] in three-dimensional space with density function [latex]\\rho(x,y,z)[\/latex] at any point [latex](x, y, z)[\/latex], the total mass is:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]m = \\iiint_Q \\rho(x,y,z) dV[\/latex]<\/p>\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">\r\n<p class=\"whitespace-normal break-words\"><strong>Two-Dimensional Mass Formula<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">For a lamina (flat plate) occupying region [latex]R[\/latex] with density [latex]\\rho(x,y)[\/latex]:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]m = \\iint_R \\rho(x,y) dA[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">The three-dimensional formula follows the same pattern, but we integrate over volume [latex]dV[\/latex] instead of area [latex]dA[\/latex].<\/p>\r\n\r\n<\/section>\r\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Moments in Three Dimensions<\/h3>\r\n<p class=\"whitespace-normal break-words\">The moments of the solid about the three coordinate planes are:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{aligned} M_{xy} = &amp; \\iiint_Q z\\rho(x,y,z) dV, \\quad M_{xz} = \\iiint_Q y\\rho(x,y,z) dV, \\ M_{yz} = &amp; \\iiint_Q x\\rho(x,y,z) dV \\end{aligned}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">Notice the pattern: the moment about each plane uses the perpendicular coordinate as the weighting factor.<\/p>\r\n\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<p class=\"whitespace-normal break-words\"><strong>Understanding Moment Subscripts<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]M_{xy}[\/latex] is the moment about the [latex]xy[\/latex]-plane, so we multiply by [latex]z[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]M_{xz}[\/latex] is the moment about the [latex]xz[\/latex]-plane, so we multiply by [latex]y[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]M_{yz}[\/latex] is the moment about the [latex]yz[\/latex]-plane, so we multiply by [latex]x[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-pre-wrap break-words\">The subscript tells you which plane, and the missing coordinate is what you multiply by.<\/p>\r\n\r\n<\/section>\r\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Center of Mass in Three Dimensions<\/h3>\r\n<p class=\"whitespace-normal break-words\">If the center of mass of the object is the point [latex](\\overline{x}, \\overline{y}, \\overline{z})[\/latex], then:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\overline{x} = \\frac{M_{yz}}{m}, \\quad \\overline{y} = \\frac{M_{xz}}{m}, \\quad \\overline{z} = \\frac{M_{xy}}{m}[\/latex]<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>center of mass for three-dimensional solids<\/h3>\r\n<p class=\"whitespace-normal break-words\">For a solid [latex]Q[\/latex] with density [latex]\\rho(x,y,z)[\/latex] and mass [latex]m[\/latex]:<\/p>\r\n\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\overline{x} = \\dfrac{M_{yz}}{m} = \\dfrac{\\iiint_Q x\\rho(x,y,z) dV}{\\iiint_Q \\rho(x,y,z) dV}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\overline{y} = \\dfrac{M_{xz}}{m} = \\dfrac{\\iiint_Q y\\rho(x,y,z) dV}{\\iiint_Q \\rho(x,y,z) dV}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\overline{z} = \\dfrac{M_{xy}}{m} = \\dfrac{\\iiint_Q z\\rho(x,y,z) dV}{\\iiint_Q \\rho(x,y,z) dV}[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">Just as with laminas, if the solid object is <strong>homogeneous<\/strong> (has constant density), then the center of mass becomes the <strong>centroid<\/strong> of the solid\u2014the geometric center independent of the density value.<\/p>\r\n\r\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Moments of Inertia in Three Dimensions<\/h3>\r\n<p class=\"whitespace-normal break-words\">The moments of inertia about the three coordinate planes are:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{aligned} I_x = &amp; \\iiint_Q (y^2 + z^2) \\rho(x,y,z) dV, \\quad I_y =\u00a0 \\iiint_Q (x^2 + z^2) \\rho(x,y,z) dV, \\quad I_z =\u00a0 \\iiint_Q (x^2 + y^2) \\rho(x,y,z) dV \\end{aligned}[\/latex]<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>moments of inertia in three dimensions<\/h3>\r\n<p class=\"whitespace-normal break-words\">For a solid [latex]Q[\/latex] with density [latex]\\rho(x,y,z)[\/latex]:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>About the [latex]x[\/latex]-axis:<\/strong> [latex]I_x = \\iiint_Q (y^2 + z^2) \\rho(x,y,z) dV[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>About the [latex]y[\/latex]-axis:<\/strong> [latex]I_y = \\iiint_Q (x^2 + z^2) \\rho(x,y,z) dV[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>About the [latex]z[\/latex]-axis:<\/strong> [latex]I_z = \\iiint_Q (x^2 + y^2) \\rho(x,y,z) dV[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-pre-wrap break-words\">Each formula uses the sum of squares of the two coordinates perpendicular to the axis.<\/p>\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<p class=\"whitespace-normal break-words\"><strong>Pattern for Moments of Inertia<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">To remember which coordinates to square:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]I_x[\/latex] uses [latex]y^2 + z^2[\/latex] (everything except [latex]x[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]I_y[\/latex] uses [latex]x^2 + z^2[\/latex] (everything except [latex]y[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]I_z[\/latex] uses [latex]x^2 + y^2[\/latex] (everything except [latex]z[\/latex])<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-pre-wrap break-words\">These represent the squared distance from each axis.<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Suppose that [latex]Q[\/latex] is a solid region bounded by [latex]x+2y+3z=6[\/latex] and the coordinate planes and has density [latex]{\\rho}{({x},{y},{z})} = {{x}^{2}}{y}{z}[\/latex]. Find the total mass.[reveal-answer q=\"875204756\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"875204756\"]\r\n<p id=\"fs-id1167793589656\">The region [latex]Q[\/latex] is a tetrahedron (Figure 1) meeting the axes at the points [latex](6, 0, 0)[\/latex], [latex](0, 3, 0)[\/latex], and [latex](0, 0, 2)[\/latex]. To find the limits of integration, let [latex]z=0[\/latex] in the slanted plane [latex]{z} = {\\frac{1}{3}}{({6} - {x} - {{2}{y}})}[\/latex]. Then for [latex]x[\/latex] and [latex]y[\/latex] find the projection of [latex]Q[\/latex] onto the [latex]xy[\/latex]-plane, which is bounded by the axes and the line [latex]x+2y=6[\/latex]. Hence the mass is<\/p>\r\n\r\n<center>\r\n[latex]\r\n{m} = {\\underset{Q}{\\displaystyle\\iiint}}{\\rho}{({x},{y},{z})}{dV} = \\displaystyle\\int^{{x} = {6}}_{{x} = {0}} \\ \\displaystyle\\int^{y = \\frac{1}{2}({6} - x)}_{y = 0} \\ \\displaystyle\\int^{z = \\frac{1}{3}(6 - x - 2y)}_{z = 0} \\ x^{2}yz \\ dz \\ dy \\ dx = \\frac{108}{35} \\ \\approx \\ 3.086.\r\n[\/latex]<\/center>[caption id=\"attachment_1414\" align=\"aligncenter\" width=\"268\"]<img class=\"size-full wp-image-1414\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/26174918\/5-6-7.jpeg\" alt=\"In x y z space, the solid Q is shown with corners (0, 0, 0), (0, 0, 2), (0, 3, 0), and (6, 0, 0). Alternatively, you could consider the solid as being bounded by the x y, x z, and y z planes and the plane x + 2y + 3z = 6, forming an irregular tetrahedron.\" width=\"268\" height=\"309\" \/> Figure 1. Finding the mass of a three-dimensional solid [latex]Q[\/latex].[\/caption][\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Suppose [latex]Q[\/latex] is a solid region bounded by the plane [latex]x+2y+3z=6[\/latex] and the coordinate planes with density [latex]{\\rho}{({x},{y},{z})} = {{x}^{2}}{y}{z}[\/latex] (see\u00a0Figure 1). Find the center of mass using decimal approximation. Use the mass found in\u00a0Example \"Finding the Mass of a Solid\".[reveal-answer q=\"723096348\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"723096348\"]\r\n<p id=\"fs-id1167793777311\">We have used this tetrahedron before and know the limits of integration, so we can proceed to the computations right away. First, we need to find the moments about the\u00a0[latex]xy[\/latex]-plane, the\u00a0[latex]xz[\/latex]-plane, and the\u00a0[latex]yz[\/latex]-plane:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\r\n{{M}_{xy}} = &amp;{\\underset{Q}{\\displaystyle\\iiint}}{z}{\\rho}{({x},{y},{z})}{dV} = {\\displaystyle\\int^{{x} = {6}}_{{x} = {0}}} \\ {\\displaystyle\\int^{{y} = {1\/2}{({6} - {x})}}_{{y} = {0}}} \\ {\\displaystyle\\int^{{z} = {1\/3}{({6} - {x} - {2y})}}_{{z} = {0}}} \\ {{x}^{2}}{{yz}^{2}}{dz} \\ {dy} \\ {dx} = {\\frac{54}{35}} \\ {\\approx} \\ {1.543}, \\\\\r\n{{M}_{xz}} = &amp; {\\underset{Q}{\\displaystyle\\iiint}}{y}{\\rho}{({x},{y},{z})}{dV} = {\\displaystyle\\int^{{x} = {6}}_{{x} = {0}}} \\ {\\displaystyle\\int^{{y} = {1\/2}{({6} - {x})}}_{{y} = {0}}} \\ {\\displaystyle\\int^{{z} = {1\/3}{({6} - {x} - {2y})}}_{{z} = {0}}} \\ {{x}^{2}}{{y}^{2}}{z} \\ {dz} \\ {dy} \\ {dx} = {\\frac{81}{35}} \\ {\\approx} \\ {2.314}, \\\\\r\n{{M}_{yz}} = &amp; {\\underset{Q}{\\displaystyle\\iiint}}{x}{\\rho}{({x},{y},{z})}{dV} = {\\displaystyle\\int^{{x} = {6}}_{{x} = {0}}} \\ {\\displaystyle\\int^{{y} = {1\/2}{({6} - {x})}}_{{y} = {0}}} \\ {\\displaystyle\\int^{{z} = {1\/3}{({6} - {x} - {2y})}}_{{z} = {0}}} \\ {{x}^{3}}{yz} \\ {dz} \\ {dy} \\ {dx} = {\\frac{243}{35}} \\ {\\approx} \\ {6.943}.\r\n\\end{aligned}[\/latex]<\/p>\r\nHence the center of mass is\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\r\n{\\overline{x}} = &amp; {\\frac{{M}_{yz}}{m}}, \\ {\\overline{y}} = {\\frac{{M}_{xz}}{m}}, \\ {\\overline{z}} = {\\frac{{M}_{xy}}{m}}, \\\\\r\n{\\overline{x}} = &amp; {\\frac{{M}_{yz}}{m}} = {\\frac{243\/35}{108\/35}} = {\\frac{243}{108}} = {2.25}, \\\\\r\n{\\overline{y}} = &amp; {\\frac{{M}_{xz}}{m}} = {\\frac{81\/35}{108\/35}} = {\\frac{81}{108}} = {0.75}, \\\\\r\n{\\overline{z}} = &amp; {\\frac{{M}_{xy}}{m}} = {\\frac{54\/35}{108\/35}} = {\\frac{54}{108}} = {0.5}.\r\n\\end{aligned}[\/latex]<\/p>\r\nThe center of mass for the tetrahedron [latex]Q[\/latex] is the point [latex]{({2.25},{0.75},{0.5})}.[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Suppose that [latex]Q[\/latex] is a solid region and is bounded by [latex]x+2y+3z=6[\/latex] and the coordinate planes with density [latex]{\\rho}{({x},{y},{z})} = {{x}^{2}}{y}{z}[\/latex] (see\u00a0Figure 1). Find the moments of inertia of the tetrahedron [latex]Q[\/latex] about the [latex]yz[\/latex]-plane, the [latex]xz[\/latex]-plane, and the [latex]xy[\/latex]-plane.[reveal-answer q=\"452110456\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"452110456\"]\r\nOnce again, we can almost immediately write the limits of integration and hence we can quickly proceed to evaluating the moments of inertia. Using the formula stated before, the moments of inertia of the tetrahedron [latex]Q[\/latex] about the [latex]xy[\/latex]-plane, the [latex]xz[\/latex]-plane, and the [latex]yz[\/latex]-plane are<center>[latex]\\begin{aligned}\r\n{{I}_{x}} = &amp; {\\underset{Q}{\\displaystyle\\iiint}}{({{y}^{2}} + {{z}^{2}})}{\\rho}{({x},{y},{z})}{dV}, \\\\\r\n{{I}_{y}} = &amp; {\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{z}^{2}})}{\\rho}{({x},{y},{z})}{dV},\r\n\\end{aligned}[\/latex]<\/center>\r\nand<center>[latex]{{I}_{z}} = {\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{y}^{2}})}{\\rho}{({x},{y},{z})}{dV} \\ {\\text{with}} \\ {\\rho}{({x},{y},{z})} = {{x}^{2}}{yz}.[\/latex]<\/center>\r\nProceeding with the computations, we have<center>[latex]\\begin{align}<\/center>{{I}_{x}} &amp;= {\\underset{Q}{\\displaystyle\\iiint}}{({{y}^{2}} + {{z}^{2}})}x^2yz \\ {dV}=\\displaystyle\\int_{x=0}^{x=6}\\displaystyle\\int_{y=0}^{y=\\frac{1}{2}(6-x)}\\displaystyle\\int_{z=0}^{z=\\frac13(6-x-2y)}(y^2+z^2)x^2yz \\ dz \\ dy \\ dx = \\frac{117}{35}\\approx3.343, \\\\{{I}_{y}} &amp;= {\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{z}^{2}})}x^2yz \\ {dV}=\\displaystyle\\int_{x=0}^{x=6}\\displaystyle\\int_{y=0}^{y=\\frac{1}{2}(6-x)}\\displaystyle\\int_{z=0}^{z=\\frac13(6-x-2y)}(x^2+z^2)x^2yz \\ dz \\ dy \\ dx = \\frac{684}{35}\\approx19.543, \\\\{{I}_{z}} &amp;= {\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{y}^{2}})}x^2yz \\ {dV}=\\displaystyle\\int_{x=0}^{x=6}\\displaystyle\\int_{y=0}^{y=\\frac{1}{2}(6-x)}\\displaystyle\\int_{z=0}^{z=\\frac13(6-x-2y)}(x^2+y^2)x^2yz \\ dz \\ dy \\ dx = \\frac{729}{35}\\approx20.829.\\end{align}\r\n\r\n[\/latex]\r\n\r\nThus, the moments of inertia of the tetrahedron [latex]Q[\/latex] about the\u00a0[latex]yz[\/latex]-plane, the\u00a0[latex]xz[\/latex]-plane, and the\u00a0[latex]xy[\/latex]-plane are [latex]{117\/35}, {684\/35},[\/latex] and [latex]{729\/35}[\/latex], respectively.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"<h2>Center of Mass and Moments of Inertia in Three Dimensions<\/h2>\n<p>All the formulas we developed for double integrals can be extended to triple integrals for three-dimensional objects.<\/p>\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Mass of a Three-Dimensional Solid<\/h3>\n<p class=\"whitespace-normal break-words\">For a solid object [latex]Q[\/latex] in three-dimensional space with density function [latex]\\rho(x,y,z)[\/latex] at any point [latex](x, y, z)[\/latex], the total mass is:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]m = \\iiint_Q \\rho(x,y,z) dV[\/latex]<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">\n<p class=\"whitespace-normal break-words\"><strong>Two-Dimensional Mass Formula<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">For a lamina (flat plate) occupying region [latex]R[\/latex] with density [latex]\\rho(x,y)[\/latex]:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]m = \\iint_R \\rho(x,y) dA[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">The three-dimensional formula follows the same pattern, but we integrate over volume [latex]dV[\/latex] instead of area [latex]dA[\/latex].<\/p>\n<\/section>\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Moments in Three Dimensions<\/h3>\n<p class=\"whitespace-normal break-words\">The moments of the solid about the three coordinate planes are:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{aligned} M_{xy} = & \\iiint_Q z\\rho(x,y,z) dV, \\quad M_{xz} = \\iiint_Q y\\rho(x,y,z) dV, \\ M_{yz} = & \\iiint_Q x\\rho(x,y,z) dV \\end{aligned}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">Notice the pattern: the moment about each plane uses the perpendicular coordinate as the weighting factor.<\/p>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<p class=\"whitespace-normal break-words\"><strong>Understanding Moment Subscripts<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]M_{xy}[\/latex] is the moment about the [latex]xy[\/latex]-plane, so we multiply by [latex]z[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]M_{xz}[\/latex] is the moment about the [latex]xz[\/latex]-plane, so we multiply by [latex]y[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]M_{yz}[\/latex] is the moment about the [latex]yz[\/latex]-plane, so we multiply by [latex]x[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-pre-wrap break-words\">The subscript tells you which plane, and the missing coordinate is what you multiply by.<\/p>\n<\/section>\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Center of Mass in Three Dimensions<\/h3>\n<p class=\"whitespace-normal break-words\">If the center of mass of the object is the point [latex](\\overline{x}, \\overline{y}, \\overline{z})[\/latex], then:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\overline{x} = \\frac{M_{yz}}{m}, \\quad \\overline{y} = \\frac{M_{xz}}{m}, \\quad \\overline{z} = \\frac{M_{xy}}{m}[\/latex]<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>center of mass for three-dimensional solids<\/h3>\n<p class=\"whitespace-normal break-words\">For a solid [latex]Q[\/latex] with density [latex]\\rho(x,y,z)[\/latex] and mass [latex]m[\/latex]:<\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]\\overline{x} = \\dfrac{M_{yz}}{m} = \\dfrac{\\iiint_Q x\\rho(x,y,z) dV}{\\iiint_Q \\rho(x,y,z) dV}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\overline{y} = \\dfrac{M_{xz}}{m} = \\dfrac{\\iiint_Q y\\rho(x,y,z) dV}{\\iiint_Q \\rho(x,y,z) dV}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\overline{z} = \\dfrac{M_{xy}}{m} = \\dfrac{\\iiint_Q z\\rho(x,y,z) dV}{\\iiint_Q \\rho(x,y,z) dV}[\/latex]<\/li>\n<\/ul>\n<\/section>\n<p class=\"whitespace-normal break-words\">Just as with laminas, if the solid object is <strong>homogeneous<\/strong> (has constant density), then the center of mass becomes the <strong>centroid<\/strong> of the solid\u2014the geometric center independent of the density value.<\/p>\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Moments of Inertia in Three Dimensions<\/h3>\n<p class=\"whitespace-normal break-words\">The moments of inertia about the three coordinate planes are:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\begin{aligned} I_x = & \\iiint_Q (y^2 + z^2) \\rho(x,y,z) dV, \\quad I_y =\u00a0 \\iiint_Q (x^2 + z^2) \\rho(x,y,z) dV, \\quad I_z =\u00a0 \\iiint_Q (x^2 + y^2) \\rho(x,y,z) dV \\end{aligned}[\/latex]<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>moments of inertia in three dimensions<\/h3>\n<p class=\"whitespace-normal break-words\">For a solid [latex]Q[\/latex] with density [latex]\\rho(x,y,z)[\/latex]:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>About the [latex]x[\/latex]-axis:<\/strong> [latex]I_x = \\iiint_Q (y^2 + z^2) \\rho(x,y,z) dV[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>About the [latex]y[\/latex]-axis:<\/strong> [latex]I_y = \\iiint_Q (x^2 + z^2) \\rho(x,y,z) dV[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>About the [latex]z[\/latex]-axis:<\/strong> [latex]I_z = \\iiint_Q (x^2 + y^2) \\rho(x,y,z) dV[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-pre-wrap break-words\">Each formula uses the sum of squares of the two coordinates perpendicular to the axis.<\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<p class=\"whitespace-normal break-words\"><strong>Pattern for Moments of Inertia<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">To remember which coordinates to square:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]I_x[\/latex] uses [latex]y^2 + z^2[\/latex] (everything except [latex]x[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">[latex]I_y[\/latex] uses [latex]x^2 + z^2[\/latex] (everything except [latex]y[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">[latex]I_z[\/latex] uses [latex]x^2 + y^2[\/latex] (everything except [latex]z[\/latex])<\/li>\n<\/ul>\n<p class=\"whitespace-pre-wrap break-words\">These represent the squared distance from each axis.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Suppose that [latex]Q[\/latex] is a solid region bounded by [latex]x+2y+3z=6[\/latex] and the coordinate planes and has density [latex]{\\rho}{({x},{y},{z})} = {{x}^{2}}{y}{z}[\/latex]. Find the total mass.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q875204756\">Show Solution<\/button><\/p>\n<div id=\"q875204756\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793589656\">The region [latex]Q[\/latex] is a tetrahedron (Figure 1) meeting the axes at the points [latex](6, 0, 0)[\/latex], [latex](0, 3, 0)[\/latex], and [latex](0, 0, 2)[\/latex]. To find the limits of integration, let [latex]z=0[\/latex] in the slanted plane [latex]{z} = {\\frac{1}{3}}{({6} - {x} - {{2}{y}})}[\/latex]. Then for [latex]x[\/latex] and [latex]y[\/latex] find the projection of [latex]Q[\/latex] onto the [latex]xy[\/latex]-plane, which is bounded by the axes and the line [latex]x+2y=6[\/latex]. Hence the mass is<\/p>\n<div style=\"text-align: center;\">\n[latex]{m} = {\\underset{Q}{\\displaystyle\\iiint}}{\\rho}{({x},{y},{z})}{dV} = \\displaystyle\\int^{{x} = {6}}_{{x} = {0}} \\ \\displaystyle\\int^{y = \\frac{1}{2}({6} - x)}_{y = 0} \\ \\displaystyle\\int^{z = \\frac{1}{3}(6 - x - 2y)}_{z = 0} \\ x^{2}yz \\ dz \\ dy \\ dx = \\frac{108}{35} \\ \\approx \\ 3.086.[\/latex]<\/div>\n<figure id=\"attachment_1414\" aria-describedby=\"caption-attachment-1414\" style=\"width: 268px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1414\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/26174918\/5-6-7.jpeg\" alt=\"In x y z space, the solid Q is shown with corners (0, 0, 0), (0, 0, 2), (0, 3, 0), and (6, 0, 0). Alternatively, you could consider the solid as being bounded by the x y, x z, and y z planes and the plane x + 2y + 3z = 6, forming an irregular tetrahedron.\" width=\"268\" height=\"309\" \/><figcaption id=\"caption-attachment-1414\" class=\"wp-caption-text\">Figure 1. Finding the mass of a three-dimensional solid [latex]Q[\/latex].<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Suppose [latex]Q[\/latex] is a solid region bounded by the plane [latex]x+2y+3z=6[\/latex] and the coordinate planes with density [latex]{\\rho}{({x},{y},{z})} = {{x}^{2}}{y}{z}[\/latex] (see\u00a0Figure 1). Find the center of mass using decimal approximation. Use the mass found in\u00a0Example &#8220;Finding the Mass of a Solid&#8221;.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q723096348\">Show Solution<\/button><\/p>\n<div id=\"q723096348\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793777311\">We have used this tetrahedron before and know the limits of integration, so we can proceed to the computations right away. First, we need to find the moments about the\u00a0[latex]xy[\/latex]-plane, the\u00a0[latex]xz[\/latex]-plane, and the\u00a0[latex]yz[\/latex]-plane:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}  {{M}_{xy}} = &{\\underset{Q}{\\displaystyle\\iiint}}{z}{\\rho}{({x},{y},{z})}{dV} = {\\displaystyle\\int^{{x} = {6}}_{{x} = {0}}} \\ {\\displaystyle\\int^{{y} = {1\/2}{({6} - {x})}}_{{y} = {0}}} \\ {\\displaystyle\\int^{{z} = {1\/3}{({6} - {x} - {2y})}}_{{z} = {0}}} \\ {{x}^{2}}{{yz}^{2}}{dz} \\ {dy} \\ {dx} = {\\frac{54}{35}} \\ {\\approx} \\ {1.543}, \\\\  {{M}_{xz}} = & {\\underset{Q}{\\displaystyle\\iiint}}{y}{\\rho}{({x},{y},{z})}{dV} = {\\displaystyle\\int^{{x} = {6}}_{{x} = {0}}} \\ {\\displaystyle\\int^{{y} = {1\/2}{({6} - {x})}}_{{y} = {0}}} \\ {\\displaystyle\\int^{{z} = {1\/3}{({6} - {x} - {2y})}}_{{z} = {0}}} \\ {{x}^{2}}{{y}^{2}}{z} \\ {dz} \\ {dy} \\ {dx} = {\\frac{81}{35}} \\ {\\approx} \\ {2.314}, \\\\  {{M}_{yz}} = & {\\underset{Q}{\\displaystyle\\iiint}}{x}{\\rho}{({x},{y},{z})}{dV} = {\\displaystyle\\int^{{x} = {6}}_{{x} = {0}}} \\ {\\displaystyle\\int^{{y} = {1\/2}{({6} - {x})}}_{{y} = {0}}} \\ {\\displaystyle\\int^{{z} = {1\/3}{({6} - {x} - {2y})}}_{{z} = {0}}} \\ {{x}^{3}}{yz} \\ {dz} \\ {dy} \\ {dx} = {\\frac{243}{35}} \\ {\\approx} \\ {6.943}.  \\end{aligned}[\/latex]<\/p>\n<p>Hence the center of mass is<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}  {\\overline{x}} = & {\\frac{{M}_{yz}}{m}}, \\ {\\overline{y}} = {\\frac{{M}_{xz}}{m}}, \\ {\\overline{z}} = {\\frac{{M}_{xy}}{m}}, \\\\  {\\overline{x}} = & {\\frac{{M}_{yz}}{m}} = {\\frac{243\/35}{108\/35}} = {\\frac{243}{108}} = {2.25}, \\\\  {\\overline{y}} = & {\\frac{{M}_{xz}}{m}} = {\\frac{81\/35}{108\/35}} = {\\frac{81}{108}} = {0.75}, \\\\  {\\overline{z}} = & {\\frac{{M}_{xy}}{m}} = {\\frac{54\/35}{108\/35}} = {\\frac{54}{108}} = {0.5}.  \\end{aligned}[\/latex]<\/p>\n<p>The center of mass for the tetrahedron [latex]Q[\/latex] is the point [latex]{({2.25},{0.75},{0.5})}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Suppose that [latex]Q[\/latex] is a solid region and is bounded by [latex]x+2y+3z=6[\/latex] and the coordinate planes with density [latex]{\\rho}{({x},{y},{z})} = {{x}^{2}}{y}{z}[\/latex] (see\u00a0Figure 1). Find the moments of inertia of the tetrahedron [latex]Q[\/latex] about the [latex]yz[\/latex]-plane, the [latex]xz[\/latex]-plane, and the [latex]xy[\/latex]-plane.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q452110456\">Show Solution<\/button><\/p>\n<div id=\"q452110456\" class=\"hidden-answer\" style=\"display: none\">\nOnce again, we can almost immediately write the limits of integration and hence we can quickly proceed to evaluating the moments of inertia. Using the formula stated before, the moments of inertia of the tetrahedron [latex]Q[\/latex] about the [latex]xy[\/latex]-plane, the [latex]xz[\/latex]-plane, and the [latex]yz[\/latex]-plane are<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{aligned}  {{I}_{x}} = & {\\underset{Q}{\\displaystyle\\iiint}}{({{y}^{2}} + {{z}^{2}})}{\\rho}{({x},{y},{z})}{dV}, \\\\  {{I}_{y}} = & {\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{z}^{2}})}{\\rho}{({x},{y},{z})}{dV},  \\end{aligned}[\/latex]<\/div>\n<p>and<\/p>\n<div style=\"text-align: center;\">[latex]{{I}_{z}} = {\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{y}^{2}})}{\\rho}{({x},{y},{z})}{dV} \\ {\\text{with}} \\ {\\rho}{({x},{y},{z})} = {{x}^{2}}{yz}.[\/latex]<\/div>\n<p>Proceeding with the computations, we have<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}<\/div>\n<p>{{I}_{x}} &= {\\underset{Q}{\\displaystyle\\iiint}}{({{y}^{2}} + {{z}^{2}})}x^2yz \\ {dV}=\\displaystyle\\int_{x=0}^{x=6}\\displaystyle\\int_{y=0}^{y=\\frac{1}{2}(6-x)}\\displaystyle\\int_{z=0}^{z=\\frac13(6-x-2y)}(y^2+z^2)x^2yz \\ dz \\ dy \\ dx = \\frac{117}{35}\\approx3.343, \\\\{{I}_{y}} &= {\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{z}^{2}})}x^2yz \\ {dV}=\\displaystyle\\int_{x=0}^{x=6}\\displaystyle\\int_{y=0}^{y=\\frac{1}{2}(6-x)}\\displaystyle\\int_{z=0}^{z=\\frac13(6-x-2y)}(x^2+z^2)x^2yz \\ dz \\ dy \\ dx = \\frac{684}{35}\\approx19.543, \\\\{{I}_{z}} &= {\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{y}^{2}})}x^2yz \\ {dV}=\\displaystyle\\int_{x=0}^{x=6}\\displaystyle\\int_{y=0}^{y=\\frac{1}{2}(6-x)}\\displaystyle\\int_{z=0}^{z=\\frac13(6-x-2y)}(x^2+y^2)x^2yz \\ dz \\ dy \\ dx = \\frac{729}{35}\\approx20.829.\\end{align}[\/latex]<\/p>\n<p>Thus, the moments of inertia of the tetrahedron [latex]Q[\/latex] about the\u00a0[latex]yz[\/latex]-plane, the\u00a0[latex]xz[\/latex]-plane, and the\u00a0[latex]xy[\/latex]-plane are [latex]{117\/35}, {684\/35},[\/latex] and [latex]{729\/35}[\/latex], respectively.<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":19,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1287","chapter","type-chapter","status-publish","hentry"],"part":434,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1287","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1287\/revisions"}],"predecessor-version":[{"id":1298,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1287\/revisions\/1298"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/434"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1287\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=1287"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=1287"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=1287"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=1287"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}